Leveraging Diversity and Sparsity in Blind Deconvolution

Ahmed, Ali; Demanet, Laurent

doi:10.1109/tit.2017.2788444

Cited by 23 publications

(34 citation statements)

References 32 publications

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“…Similar coherence parameters appear in the related recent literature on blind deconvolution [10,13], and elsewhere in compressed sensing [15,16], in general. Without loss of generality, we assume that h 0 2 = √ d 0 , and x 0 2 = √ d 0 .…”

Section: Coherence Parameterssupporting

confidence: 69%

Channel Protection Using Random Modulation

Ahmed

Hameed

2019

ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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This paper shows that modulation protects a bandlimited signal against convolutive interference. A signal s(t), bandlimited to BHz, is modulated (pointwise multiplied) with a known random sign sequence r(t), alternating at a rate Q, and the resultant spread spectrum signal s(t) r(t) is convolved against an M-tap channel impulse response h(t) to yield the observed signal y(t) = (s(t) r(t)) h(t), where and denote pointwise multiplication, and circular convolution, respectively.We show that both s(t), and h(t) can be provably recovered using a simple gradient descent scheme by alternating the binary waveform r(t) at a rate Q B+M(to within log factors and a signal coherences) and sampling y(t) at a rate Q. We also present a comprehensive set of phase transitions to depict the trade-off between Q, M, and B for successful recovery. Moreover, we show stable recovery results under noise.

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Section: Coherence Parameterssupporting

confidence: 69%

Channel Protection Using Random Modulation

Ahmed

Hameed

2019

ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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“…where as noted in the notations section above, C ⊗ N is a block diagonal matrix formed by stacking together N matrices C. Similar coherence parameters appear in the related recent literature on blind deconvolution [9], [12], and elsewhere in compressed sensing [14], [15], in general. Without loss of generality, we only assume that h 0 2 = √ d 0 , and x 0 2 = √ d 0 .…”

Section: B Coherence Parametersmentioning

confidence: 70%

“…Additionally, we take h to be completely arbitrary, and do not assume any structure such as a known subspace or sparsity in a known basis; this is unlike some of the other recent works [9]- [12]. In general, we take M ≤ L as already assumed in the observation model (1).…”

Section: Introductionmentioning

confidence: 99%

Blind Deconvolution Using Modulated Inputs

Ahmed

2020

IEEE Trans. Signal Process.

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This paper considers the blind deconvolution of multiple modulated signals/filters, and an arbitrary filter/signal. Multiple inputs s 1 , s 2 , . . . , s N =: [s n ] are modulated (pointwise multiplied) with random sign sequences r 1 , r 2 , . . . , r N =: [r n ], respectively, and the resultant inputs (s n r n ) ∈ C Q , n = [N] are convolved against an arbitrary input h ∈ C M to yield the measurements y n = (s n r n ) h, n = [N] := 1, 2, . . . , N, where , and denote pointwise multiplication, and circular convolution. Given [y n ], we want to recover the unknowns [s n ], and h. We make a structural assumption that unknown [s n ] are members of a known K-dimensional (not necessarily random) subspace, and prove that the unknowns can be recovered from sufficiently many observations using an alternating gradient descent algorithm whenever the modulated inputs s n r n are long enough, i.e, Q K N+M (to within log factors and signal dispersion/coherence parameters).To the best of our knowledge, this is the first provable result of multichannel blind deconvolution using gradient descent, and importantly under comparatively lenient structural assumptions on the convolved inputs. A neat conclusion drawn from this result is that modulation of a bandlimited signal protects it against an unknown convolutive distortion. We discuss the applications of this result in passive imaging, wireless communication in unknown environment, and image deblurring. A thorough numerical investigation of the theoretical results is also presented using phase transitions, image deblurring experiments, and noise stability plots.

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“…Another related bilinear inverse problem is blind calibration via repeated measurements from multiple sensing operators [33], [34], [35], [36], [37], [38]. Since blind calibration with repeated measurements is in principle an easier problem than BGPC [7], we believe our methods for BGPC and our theoretical analysis can be extended to this scenario.…”

Section: Related Workmentioning

confidence: 99%

Blind Gain and Phase Calibration via Sparse Spectral Methods

Lee

Bresler

2019

IEEE Trans. Inform. Theory

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Blind gain and phase calibration (BGPC) is a bilinear inverse problem involving the determination of unknown gains and phases of the sensing system, and the unknown signal, jointly. BGPC arises in numerous applications, e.g., blind albedo estimation in inverse rendering, synthetic aperture radar autofocus, and sensor array auto-calibration. In some cases, sparse structure in the unknown signal alleviates the ill-posedness of BGPC. Recently there has been renewed interest in solutions to BGPC with careful analysis of error bounds. In this paper, we formulate BGPC as an eigenvalue/eigenvector problem, and propose to solve it via power iteration, or in the sparsity or joint sparsity case, via truncated power iteration. Under certain assumptions, the unknown gains, phases, and the unknown signal can be recovered simultaneously. Numerical experiments show that power iteration algorithms work not only in the regime predicted by our main results, but also in regimes where theoretical analysis is limited. We also show that our power iteration algorithms for BGPC compare favorably with competing algorithms in adversarial conditions, e.g., with noisy measurement or with a bad initial estimate.

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Leveraging Diversity and Sparsity in Blind Deconvolution

Cited by 23 publications

References 32 publications

Channel Protection Using Random Modulation

Channel Protection Using Random Modulation

Blind Deconvolution Using Modulated Inputs

Blind Gain and Phase Calibration via Sparse Spectral Methods

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